mesh simplification - computer...

Mesh Simplification

ApplicationsApplications

• Oversampled 3D scan data• Oversampled 3D scan data

~150k triangles ~80k triangles

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• Overtessellation: E g iso surface extraction• Overtessellation: E.g. iso-surface extraction

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• Multi resolution hierarchies for• Multi-resolution hierarchies for – efficient geometry processing– level-of-detail (LOD) rendering

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• Adaptation to hardware capabilities• Adaptation to hardware capabilities

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Size-Quality TradeoffSize Quality Tradeoff

errorerror

size

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size

Problem StatementProblem Statement

• Given:• Given:

• Find: such that

1. and is minimal, or

2. and is minimal

• Respect additional fairness criteria– normal deviation, triangle shape, scalar attributes,

etc.

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Mesh Decimation MethodsMesh Decimation Methods

• Vertex clustering• Vertex clustering

• Incremental decimation

Resampling• Resampling

• Mesh approximation

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Vertex ClusteringVertex Clustering

• Cluster Generation• Cluster Generation

• Computing a representativep g p

• Mesh generation

• Topology changes

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• Cluster Generation• Cluster Generation– Uniform 3D grid– Map vertices to cluster cells

• Computing a representative• Computing a representative

• Mesh generation


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• Cluster Generation• Cluster Generation– Hierarchical approach– Top-down or bottom-up

• Computing a representative• Computing a representative

• Mesh generation


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• Computing a representativep g p– Average/median vertex position

Error quadrics– Error quadrics

• Mesh generation


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Computing a RepresentativeComputing a Representative

Average vertex position

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Median vertex position

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Error quadrics

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Error QuadricsError Quadrics

• Patch is expected to be piecewise flat• Patch is expected to be piecewise flat

• Minimize distance to neighboring triangles’ planesg g g p

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Error QuadricsError Quadrics

• Squared distance of point p to plane q:• Squared distance of point p to plane q:

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Error Quadrics

• Sum distances to planes q of vertex’

Error Quadrics

• Sum distances to planes qi of vertex neighboring triangles:

• Point p* that minimizes the error satisfies:Point p that minimizes the error satisfies:

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ComparisonComparison

d idi error quadricaverage median

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• Mesh generation– Clusters p ↔ {p0,...,pn}, q ↔ {q0,...,qm}

• Topology changesTopology changes

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• Mesh generation– Clusters p ↔ {p0,...,pn}, q ↔ {q0,...,qm} – Connect (p,q) if there was an edge (pi,qj)j


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• Mesh generation

• Topology changes– If different sheets passIf different sheets pass

through one cellCan be non manifold– Can be non-manifold

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OutlineOutline

• Applications• Applications

• Problem Statement

• Mesh Decimation Methods– Vertex Clustering– Incremental Decimation– Extensions

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Incremental DecimationIncremental Decimation

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• General Setup• General Setup

• Decimation operatorsp

• Error metrics

• Fairness criteria

Topolog changes• Topology changes

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General SetupGeneral Setup

• Repeat:

• pick mesh region

• apply decimation operatorpp y p

• Until no further reduction possible

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Greedy OptimizationGreedy Optimization• For each regionFor each region• evaluate quality after decimation• enqeue(quality region)• enqeue(quality, region)

• Repeat:• get best mesh region from queue• apply decimation operator• update queuep q• Until no further reduction possible

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Global Error ControlGlobal Error Control• For each regionFor each region• evaluate quality after decimation• enqeue(quality region)• enqeue(quality, region)

• Repeat:• get best mesh region from queue• if error < ε• apply decimation operatorpp y p• update queue• Until no further reduction possible

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Until no further reduction possible




• Error metrics



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Decimation OperatorsDecimation Operators

• What is a "region" ?• What is a region ?

• What are the DOF for re-triangulation?g

• Classification– Topology-changing vs. topology-preserving– Subsampling vs. filtering– Inverse operation → progressive meshes

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Vertex RemovalVertex Removal

Select a vertex to be eliminatedbe eliminated

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Select all triangles sharing this vertexsharing this vertex

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Remove the selected trianglesselected triangles, creating the hole

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Fill the hole with new trianglesnew triangles

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Vertex Removal

V t I tiVertex Insertion

• Remove vertex

R t i l t h l• Re-triangulate hole– Combinatorial degrees of freedom

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Vertex Split

Edge Collapse

Vertex Split

• Merge two adjacent vertices

• Define new vertex position– Continuous degrees of freedomg– Filter along the way

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Half-Edge Collapse

C ll d i t d i t

Restricted Vertex Split

• Collapse edge into one end point– Special case of vertex removal– Special case of edge collapse

• No degrees of freedomNo degrees of freedom• Separates global optimization from local

optimization37

optimization

Half-Edge CollapseHalf Edge Collapse

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!flip

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• Error metrics



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Local Error Metrics

• Local distance to mesh

Local Error Metrics

• Local distance to mesh– Compute average plane– No comparison to original geometry

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Global Error MetricsGlobal Error Metrics

• Error quadrics• Error quadrics– Squared distance to planes at vertex– No bound on true error

Q3 = Q1+Q2Q3 Q1+Q2

p

Q2

Q1v3

solve v3TQ3v3 = min

< ? k

p1 p2

piTQipi = 0, i={1,2}

< ε ? → ok

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• Error metrics



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Fairness Criteria

• Rate quality of decimation operation

Fairness Criteria

• Rate quality of decimation operation– Approximation error– Triangle shape– Dihedral anglesg– Valence balance– ...

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Fairness Criteria

• Rate quality after decimation

Fairness Criteria

• Rate quality after decimation– Approximation error

r1

– Triangle shape– Dihedral angles

1

g– Valence balance– ...

r2

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Fairness Criteria


Fairness Criteria

• Rate quality after decimation– Approximation error– Triangle shape– Dihedral anglesg– Valence balance– ...

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Fairness Criteria


Fairness Criteria

• Rate quality after decimation– Approximation error– Triangle shape– Dihedral anglesg– Valence balance

Color differences– Color differences– ...

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Fairness Criteria


Fairness Criteria



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Fairness Criteria


Fairness Criteria



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Fairness Criteria


Fairness Criteria



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• Error metrics



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Topology Changes ?

• Merge vertices across non edges

Topology Changes ?

• Merge vertices across non-edges– Changes mesh topology– Need spatial neighborhood information– Generates non-manifold meshes

Vertex ContractionVertex Contraction

Vertex Separation

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Topology Changes ?

• Merge vertices across non edges

Topology Changes ?

• Merge vertices across non-edges– Changes mesh topology– Need spatial neighborhood information– Generates non-manifold meshes

manifold non-manifold

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ComparisonComparison

• Vertex clustering• Vertex clustering– fast, but difficult to control simplified mesh– topology changes, non-manifold meshes– global error bound, but often not close to optimumg , p

• Incremental decimation with quadric error metrics – good trade-off between mesh quality and speed– explicit control over mesh topology– restricting normal deviation improves mesh quality

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